I have recently done something similar to this using the "Haversine Formula" on WGS-84 data, which is a derivative of the "Law of Haversines" with very satisfying results.
Yes, WGS-84 assumes the Earth is an ellipsoid, but I believe you only get about a 0.5% average error using an approach like the "Haversine Formula", which may be an acceptable amount of error in your case. You will always have some amount of error unless you're talking about a distance of a few feet and even then there is theoretically curvature of the Earth... If you require more rigidly WGS-84 compatible approach checkout the "Vincenty Formula."
I understand where starblue is coming from, but good software engineering is often about trade offs, so it all depends on the accuracy you require for what you are doing. For example, the result calculated from "Manhattan Distance Formula" versus the result from the "Distance Formula" can be better for certain situations as it is computationally less expensive. Think "which point is closest?" scenarios where you don't need a precise distance measurement.
Regarding, the "Haversine Formula" it is easy to implement and is nice because it is uses "Spherical Trigonometry" instead of a "Law of Cosines" based approach which is based on two-dimensional trigonometry, therefore you get a nice balance of accuracy over complexity.
A gentlemen by the name of Chris Veness has a great website at http://www.movable-type.co.uk/scripts/latlong.html that explains some the concepts you are interested in and demonstrates various programmatic implementations; this should answer your x/y conversion question as well.
answered Jul 26 '09 at 21:04
You can do this by converting lat/long/alt into XYZ format for both points. Then, figure out the rotation angles to move one of those points (usually, the oldest point) so that it would be at lat=0 long=0 alt=0. Rotate the second position report (the newest point) by the same rotation angles. If you do it all correctly, you will find X equals the east offset, Y equals the north offset, and Z equals the up offset. You can use Pythagorean theorm with X and Y (north and east) offsets to determine the horizontal distance traveled. Normally, you just ignore the altitude differences and work with horizontal data only.
All of this assumes you are using accurate formulas to convert lat/lon/alt into XYZ. It also assumes you have enough precision in the lat/lon/alt values to be accurate. Approximations are not good if you want good results. Normally, you need about 6 decimal digits of precision in lat/lon values to compute positions down to the meter level of accuracy.
Keep in mind that this method doesn't work very well if you haven't moved far (greater than about 10 or 20 meters, more is better). There is enough noise in the GPS position reports that you are going to get jumpy velocity values that you will need to further filter to get good accuracy. The math approach isn't the problem here, it's the inherent noise in the GPS position reports. When you have good reports, you will get good velocity.
A GPS receiver doesn't normally use this approach to know velocity. It looks at the way doppler values change for each satellite and factor in current position to know what the velocity is. This works reasonably well when the vehicle is moving. It is a much faster way to detect changes in velocity (for instance, to release a position clamp). The normal user doesn't have access to the internal doppler values and the math gets very complicated, so it's not something you can do.